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Spectral responsivity interpolation of silicon CCDs
A. Ferrero, J. Campos and A. Pons
Instituto de Física Aplicada, CSIC, Madrid, Spain
Abstract. An interpolation procedure for CCD
responsivity that takes into account physical considerations
about the pixel is presented. The procedure has been
applied to the responsivity values of a CCD and the results
are shown.
Introduction
After the radiometric calibration of any detector, it may be
necessary to interpolate in order to estimate responsivity
values between the calibration wavelengths. A typical
spectral responsivity feature of a front illuminated CCD
cannot be accurately approximated by a single
mathematical function because of its shape. Then a
different approach must be followed to interpolate.
The spectral responsivity has to be based on the
absorption and charge collection in the pixels, therefore an
interpolation procedure, based on estimating the pixels’
reflectance and internal quantum efficiency, is proposed in
this work. Such an interpolation allows to predict
responsivity changes in changing the parameters and vice
versa. Its feasibility has been proved by applying it to
experimental responsivity values of a CCD calibrated in
our laboratory, whose results have been presented in
previous works. 1,2
Pixel responsivity
According to the international vocabulary of
Illumination 3 , the responsivity of a pixel (R) is the ratio
between the response and the radiant exposure. So, R can
be written as:
R
()
i
=
() 1
()
)
(
0
Khc
Where h is the Planck´s constant, c is the light´s velocity,
K is the conversion factor e - /counts, 0 is the vacuum
wavelength, is the pixel reflectance, A is the pixel
sense area, and i is the pixel internal quantum efficiency.
This last quantity can be expressed for a front-illuminated
CCD as 4 :
=
i
opt
exp
where is the silicon absorption coefficient, W is the
depletion region depth, X PSi is the width of the polisilicon
layer, opt is the optical efficiency (that accounts the
responsivity variation due to the incidence angle variation),
and is a factor that represents the non ideality of the
charge collection and charge transfer processes, whose
value is close to 1.
Interpolation procedure
A
() 1
( ) [ 1
exp( W)
] (2)
X PSi
The interpolation procedure consists of fitting equation 1
to the experimental spectral responsivity values, using
equation 2 for the internal quantum efficiency. The
variables involved in the fitting would be A, K, , opt , ,
X PSi and W. The absorption coefficient and physical
constants values can be obtained from the literature. opt
can be taken as unity for the purpose of this work. In a first
approach and to simplify the fitting, the discussion can be
restricted to scientific CCD, then, A can be considered as a
constant for all the pixels and a typical value given by the
CCD manufacturer can be taken for it as well as for K.
can also be assumed to be unity for this kind of CCD and
therefore the fitting variables are reduced to , X PSi and W.
X PSi and W can be assumed to be equal for all the pixels,
but the reflectance may change from one to another in
addition of being a function of wavelength, radiation solid
angle, polarization, etc.
Since is difficult to measure because the pixels are
very small and difficult to predict because the CCD is a
very complex system from the reflectance point of view,
an iterative method can be used to obtain physically
significant reflectance values:
1. From typical values for W and X PSi spectral quantum
efficiency values can be calculated for every pixel (eq.
2) at the wavelengths the responsivity is known.
2. From those values, spectral reflectance can be
calculated from equation 1. If those values are
physically significant, they are taken as the actual
reflectance values, if not, new values are calculated
from a different set of W and X PSi .
To conclude the interpolation of the responsivity, it is
necessary to interpolate the reflectance values obtained
previously. Unlike the case of a single detector, the CCD
responsivity interpolation has to take into account the large
amount of pixels involved. So it is important for the
interpolation procedure to be simple and because of that a
linear interpolation of is done.
The result of this interpolation procedure is a
responsivity expression for each pixel, based on pixel’s the
physical parameters.
Reflectance
0.48
0.46
0.44
0.42
0.40
0.38
0.36
0.34
0.32
450 500 550 600 650 700
Wavelength (nm)
Figure 1: Calculated for the average responsivity of a CCD.
Proceedings NEWRAD, 17-19 October 2005, Davos, Switzerland 129